Variational Derivative for Buckling Loads of Beams and Frames

The finite-difference method is a numerical technique for obtaining approximate solutions to differential equations. The main objective of the present study is to give a new aspect to the finite-difference method by using a variational derivative. By applying this formulation, accurate values of the buckling loads of beams and frames with various end supports are obtained. The performance of this formulation is verified by comparison with numerical examples in the literature __________ Published in Prikladnaya Mekhanika, Vol. 41, No. 7, pp. 139–144, July 2005.     

Monotone Nonincreasing Random Fields on Partially Ordered Sets. I

For an arbitrary poset H and measure ρ on H × _R (where _R is the real axis), we construct a monotone decreasing stochastic field η_ρ and compute its finite-dimensional distributions. In the case where H is a Λ-semilattice and the measure ρ satisfies additional conditions, we compute various characteristics of the field η_ρ such as the expectation of the field value at a point, variance of the field value at a point, and correlation function of the field. The described construction of random fields gives a new method for constructing positive definite functions on posets. Bibliography: 6 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 301, 2003, pp. 92–143.     

Equilibrium Shapes of Pressure Vessel Domes in Asymmetric Winding

Errors in the programs of winding pressure vessels can lead to an asymmetric reinforcement structure, which affects the equilibrium shape of their domes and the tension in the left and right families of filaments. In this paper, equations determining the equilibrium shape of the domes, the tension in filaments, and the shear stresses between layers are obtained. Examples with winding trajectories in the form of geodesics and constant-deviation lines are considered. It is found that, for pressure vessels, in the absence of external torque, the tension levels in filaments of the left and right families differ considerably. It is also revealed that, for actual friction coefficients in winding, the shear stresses taken up by the binder are insignificant. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 41, No. 6, pp. 743–752, November–December, 2005.     

Puzzles in astrophysics in the past and present

About 400 years have passed since the great discoveries by Galileo, Kepler, and Newton, but astronomy still remains an important source of discoveries in physics. They start with puzzles, with phenomena difficult to explain, and phenomena which in fact need new physics for explanation. Do such puzzles exist now? There are at least three candidates: absence of absorption of TeV gamma radiation in extragalactic space (violation of Lorentz invariance?), absence of GZK cutoff in the spectrum of ultrahigh-energy cosmic rays (new particle physics?), tremendous energy (up to 10⁵⁴ erg) released in gamma ray bursts on a time scale of a second (collapsing stars or sources of a new type?). Do these puzzles really exist? A critical review of these phenomena is given.     

Order restricted randomized designs for control versus treatment comparison

We develop a method of randomizing units to treatments that relies on subjective judgement or on possible coarse modeling to produce restrictions on the randomization. The procedure thus fits within the general framework of ranked set sampling. However, instead of selecting a single unit from each set for full measurement, all units within a set are used. The units within a set are assigned to different treatments. Such an assignment translates the positive dependence among units within a set into a reduction in variation of contrasting features of the treatments. A test for treatment versus control comparison, with controlled familywise error rate, is developed along with the associated confidence intervals. The new procedure is shown to be superior to corresponding procedures based on completely randomized or ranked set sample designs. The superiority appears both in asymptotic relative efficiency and in power for finite sample sizes. Importantly, this test does not rely on perfect rankings; rather, the information in the data on the quality of rankings is exploited to maintain the level of the test when rankings are imperfect. The asymptotic relative efficiency of the test is not affected by estimation of the quality of rankings, and the finite sample performance is only mildly affected.     

Thermal conductivity of crystalline chrysotile asbestos

The thermal conductivity of crystalline chrysotile asbestos made up of hollow tubular Mg₃Si₂O₅(OH)₄ filaments is measured in the range 5–300 K. The paper discusses the possibility of using this material in studies of the thermal conductivity of thin filaments of metals and semiconductors incorporated into the channels of crystalline chrysotile asbestos tubes.     

On Bayes and unbiased estimators of loss

We consider estimation of loss for generalized Bayes or pseudo-Bayes estimators of a multivariate normal mean vector, θ. In 3 and higher dimensions, the MLEX is UMVUE and minimax but is inadmissible. It is dominated by the James-Stein estimator and by many others. Johnstone (1988, On inadmissibility of some unbiased estimates of loss,Statistical Decision Theory and Related Topics, IV (eds. S. S. Gupta and J. O. Berger), Vol. 1, 361–379, Springer, New York) considered the estimation of loss for the usual estimatorX and the James-Stein estimator. He found improvements over the Stein unbiased estimator of risk. In this paper, for a generalized Bayes point estimator of θ, we compare generalized Bayes estimators to unbiased estimators of loss. We find, somewhat surprisingly, that the unbiased estimator often dominates the corresponding generalized Bayes estimator of loss for priors which give minimax estimators in the original point estimation problem. In particular, we give a class of priors for which the generalized Bayes estimator of θ is admissible and minimax but for which the unbiased estimator of loss dominates the generalized Bayes estimator of loss. We also give a general inadmissibility result for a generalized Bayes estimator of loss. Research supported by NSF Grant DMS-97-04524.